Mix ground beef with onion, Worcestershire, salt and pepper to taste, eggs and bread crumbs. Form into 4 oval patties. Heat half the butter in a skillet over medium heat and cook the patties until nicely browned, about 6 minutes per side. Place in single layer in baking dish.

Saute mushrooms in remaining butter until nicely browned, about 5 minutes on medium-high heat. Sprinkle the flour over the mushrooms, and very gradually add the stock, whisking until it is thick, 3 to 5 minutes. Pour the mushrooms and gravy over the beef patties and bake until heated through, about 15 minutes.